Simplify the following expression: $ r = \dfrac{-1}{3} - \dfrac{-8x}{-5x - 2} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{-5x - 2}{-5x - 2}$ $ \dfrac{-1}{3} \times \dfrac{-5x - 2}{-5x - 2} = \dfrac{5x + 2}{-15x - 6} $ Multiply the second expression by $\dfrac{3}{3}$ $ \dfrac{-8x}{-5x - 2} \times \dfrac{3}{3} = \dfrac{-24x}{-15x - 6} $ Therefore $ r = \dfrac{5x + 2}{-15x - 6} - \dfrac{-24x}{-15x - 6} $ Now the expressions have the same denominator we can simply subtract the numerators: $r = \dfrac{5x + 2 + 24x }{-15x - 6} $ Distribute the negative sign: $r = \dfrac{5x + 2 + 24x}{-15x - 6}$ $r = \dfrac{29x + 2}{-15x - 6}$ Simplify the expression by dividing the numerator and denominator by -1: $r = \dfrac{-29x - 2}{15x + 6}$